International Macroeconomics & Sovereign Risk
Krugman Currency Crisis & Shadow Exchange Rate Formula
Calculate Paul Krugman’s (1979) and Flood-Garber’s (1984) shadow floating exchange rate s_tilde(0), speculative attack timing T* = (R_0 - R_min) / mu - alpha, and discrete reserve wipeout alpha * mu.
Why currency pegs collapse BEFORE central-bank foreign reserves run dry!
Policymakers defending a fixed exchange-rate peg s_bar while monetizing fiscal deficits at domestic-credit expansion rate mu > 0 often assume their peg is safe until Excess Foreign Reserves (R_0 - R_min) hit zero at Naive Horizon T_naive = (R_0 - R_min) / mu! Paul Krugman (1979) and Robert Flood & Peter Garber (1984) proved why forward-looking markets launch a sudden Speculative Attack MUCH earlier: once reserves hit floor R_min, money supply grows at rate mu, creating expected post-collapse depreciation mu and shrinking real money demand by semi-elasticity wedge alpha * mu! Defining Today’s Shadow Floating Exchange Rate s_tilde(0) = s_bar - (R_0 - R_min) + alpha * mu (the rate that would prevail if the peg collapsed today), arbitrageurs prevent a predictable capital-loss jump by wiping out the final Delta_R_attack = alpha * mu of reserves in a single instant the moment s_tilde(T*) = s_bar — at exact Attack Time T* = MAX(0, ((R_0 - R_min) / mu) - alpha)!
Verify on Fixed_Peg_Index s_bar = 4.0000 (B2), Initial_Reserves R_0 = 0.5000 (B3), Reserve_Floor R_min = 0.1000 (B4), Credit_Growth_Drift mu = 0.0800 / yr (B5), and Money-Demand_Semi_Elasticity alpha = 1.50 yr (B6): Naive Non-Speculative Exhaustion takes (0.50 - 0.10) / 0.08 = 5.00 years; Today’s Shadow Floating Rate is 4.00 - 0.40 + (1.50 * 0.08) = 3.7200 (< 4.00, so no immediate attack today); Final Sudden Reserve Wipeout is alpha * mu = 0.1200; and Exact Speculative Attack Time arrives at T* = 5.00 - 1.50 = 3.50 years (1.50 full years before naive reserve exhaustion, while reserves still look comfortable at R(T*) = 0.10 + 0.12 = 0.2200!)!
| Krugman-Flood-Garber Crisis Metric (s_bar=4.0, R_0=0.50, R_min=0.10, mu=0.08, alpha=1.5) | Formula | Verified Output |
|---|---|---|
| Today’s Shadow Floating Rate s_tilde(0) | s_bar - (R_0 - R_min) + alpha * mu | 3.7200 (< 4.00 Peg Survives Today) |
| Naive Linear Exhaustion Horizon | (R_0 - R_min) / mu = 0.40 / 0.08 | 5.00 years |
| Speculative Attack Reserve Wipeout | alpha * mu = 1.50 * 0.08 | 0.1200 Instant Reserve Drop |
| Rational Speculative Attack Timing (T*) | MAX(0, (R_0 - R_min)/mu - alpha) | 3.50 years (Attacks at R = 0.22!) |
Flood-Garber Shadow Floating Exchange Rate s_tilde(0) Formula
Whenever s_tilde(0) >= s_bar (excess reserves R_0 - R_min <= alpha * mu), speculators earn an instant riskless profit by attacking immediately at t = 0!
=LET(s_pg, B2, r_0, B3, r_mn, B4, mu_cr, B5, al_md, B6, IF(OR(r_mn < 0, r_0 < r_mn, mu_cr <= 0, al_md < 0), "Invalid Crisis Input", ROUND(s_pg - (r_0 - r_mn) + al_md * mu_cr, 4)))Exact Speculative Attack Timing T* Formula
Clamp at MAX(0, ...) so an already-unsustainable reserve stock (R_0 - R_min <= alpha * mu) reports T* = 0.00 years (immediate collapse).
=LET(r_0, B3, r_mn, B4, mu_cr, B5, al_md, B6, IF(OR(r_mn < 0, r_0 < r_mn, mu_cr <= 0, al_md < 0), "Invalid Crisis Input", ROUND(MAX(0, ((r_0 - r_mn) / mu_cr) - al_md), 4)))Try the example
Use the original values or change them to test your understanding.